Loci

This section covers Loci within Geometry and Measures.

A locus is a set of points satisfying a certain condition. For example, the locus of points that are 1cm from the origin is a circle of radius 1cm centred on the origin, since all points on this circle are 1cm from the origin.

N.B. if a point P is ‘equidistant’ from two points A and B, then the distance between P and A is the same as the distance between P and B, as illustrated here:

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The points on the line are equidistant from A and B

Don’t let the term 'locus' put you off. Questions on loci (which is the plural of locus) often don’t use the term.

Example

The diagram shows two points P and Q. On the diagram shade the region which contains all the points which satisfy both the following: the distance from P is less than 3cm, the distance from P is greater than the distance from Q.

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All of the points on the circumference of the circle are 3cm from P. Therefore all of the points satisfying the condition that the distance from P is less than 3cm are in the circle.

If we draw a line in the middle of P and Q, all of the points on this line will be the same distance from P as they are from Q. They will be therefore closer to Q, and further away from P, if they are on the right of such a line.

Therefore all of the points satisfying both of these conditions are shaded in red.

Three important loci

The word locus describes the position of points which obey a certain rule.

Three important loci are:

• The circle - the locus of points which are equidistant from a fixed point, the centre.
• The perpendicular bisector - the locus of points which are equidistant from two fixed points A and B.
• The angle bisector - the locus of points which are equidistant from two fixed lines.

Example:

The diagram shows the walls of a rectangular shed, ABCD, measuring 8m by 5m. A goat is tied to the corner of C by a rope 6m long.

The shaded area shows the part the goat can reach.

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